A Study of Clight and its Semantics

coq

CompCert is a certified C compiler which comes with a proof of semantics preservation. What this means is the following: the semantics of the C code you write is preserved by CompCert compilation passes up to the generated machine code.

I had been interested in CompCert for quite some times, and ultimately challenged myself to study Clight and its semantics. This write-up is the result of this challenge, written as I was progressing.

Installing CompCert

CompCert has been added to opam, and as a consequence can be very easily used as a library for other Coq developments. A typical use case is for a project to produce Clight (the high-level AST of CompCert), and to benefit from CompCert proofs after that.

Installing CompCert is as easy as

opam install coq-compcert

More precisely, this article uses coq-compcert.3.8.

Once opam terminates, the compcert namespace becomes available. In addition, several binaries are now available if you have correctly set your $PATH environment variable. For instance, clightgen takes a C file as an argument, and generates a Coq file which contains the Clight generated by CompCert.

Problem Statement

Our goal for this first write-up is to prove that the C function

int add (int x, int y) {
    return x + y;
}

returns the expected result, that is x + y. The clightgen tool generates (among other things) the following ASTIt has been modified in order to improve its readability. .

From compcert Require Import Clight Ctypes Clightdefs AST
                             Coqlib Cop.

Definition _x : ident := 1%positive.
Definition _y : ident := 2%positive.

Definition f_add : function :=
  {| fn_return := tint
   ; fn_callconv := cc_default
   ; fn_params := [(_x, tint); (_y, tint)]
   ; fn_vars := []
   ; fn_temps := []
   ; fn_body := Sreturn
                  (Some (Ebinop Oadd
                                (Etempvar _x tint)
                                (Etempvar _y tint)
                                tint))
  |}.

The fields of the function type are pretty self-explanatory (as it is often the case in CompCert’s ASTs as far as I can tell for now).

Identifiers in Clight are (positive) indices. The fn_body field is of type statement, with the particular constructor Sreturn whose argument is of type option expr, and statement and expr look like the two main types to study. The predicates step1 and step2 allow for reasoning about the execution of a function, step-by-step (hence the name). It appears that clightgen generates Clight terms using the function call convention encoded by step2. To reason about a complete execution, it appears that we can use star (from the Smallstep module) which is basically a trace of step. These semantics are defined as predicates (that is, they live in Prop). They allow for reasoning about state transformation, where a state is either

A function call, with a given list of arguments and a continuation
A function return, with a result and a continuation
A statement execution within a function

We import several CompCert modules to manipulate values (in our case, bounded integers).

From compcert Require Import Values Integers.
Import Int.

Putting everything together, the lemma we want to prove about f_add is the following.

Lemma f_add_spec (env : genv)
    (t : Events.trace)
    (m m' : Memory.Mem.mem)
    (v : val) (x y z : int)
    (trace : Smallstep.star step2 env
               (Callstate (Ctypes.Internal f_add)
                          [Vint x; Vint y]
                          Kstop
                          m)
               t
               (Returnstate (Vint z) Kstop m'))
  : z = add x y.

Proof Walkthrough

We introduce a custom inversion tactic which does some clean-up in addition to just perform the inversion.

Ltac smart_inv H :=
  inversion H; subst; cbn in *; clear H.

We can now try to prove our lemma.

Proof.

We first destruct trace, and we rename the generated hypothesis in order to improve the readability of these notes.

  smart_inv trace.
  rename H into Hstep.
  rename H0 into Hstar.

This generates two hypotheses.

Hstep : step1
          env
          (Callstate (Ctypes.Internal f_add)
                     [Vint x; Vint y]
                     Kstop
                     m)
          t1
          s2
Hstar : Smallstep.star
          step2
          env
          s2
          t2
          (Returnstate (Vint z) Kstop m')

In other words, to “go” from a Callstate of f_add to a Returnstate, there is a first step from a Callstate to a state s2, then a succession of steps to go from s2 to a Returnstate.

We consider the single step, in order to determine the actual value of s2 (among other things). To do that, we use smart_inv on Hstep, and again perform some renaming.

  smart_inv Hstep.
  rename le into tmp_env.
  rename e into c_env.
  rename H5 into f_entry.

This produces two effects. First, a new hypothesis is added to the context.

f_entry : function_entry1
            env
            f_add
            [Vint x; Vint y]
            m
            c_env
            tmp_env
            m1

Then, the Hstar hypothesis has been updated, because we now have a more precise value of s2. More precisely, s2 has become

State
  f_add
  (Sreturn
    (Some (Ebinop Oadd
                  (Etempvar _x tint)
                  (Etempvar _y tint)
                  tint)))
  Kstop
  c_env
  tmp_env
  m1

Using the same approach as before, we can uncover the next step.

  smart_inv Hstar.
  rename H into Hstep.
  rename H0 into Hstar.

The resulting hypotheses are

Hstep : step2 env
          (State
             f_add
             (Sreturn
               (Some
                 (Ebinop Oadd
                 (Etempvar _x tint)
                 (Etempvar _y tint)
                 tint)))
             Kstop c_env tmp_env m1) t1 s2
Hstar : Smallstep.star
          step2
          env
          s2
          t0
          (Returnstate (Vint z) Kstop m')

An inversion of Hstep can be used to learn more about its resulting state… So let’s do just that.

  smart_inv Hstep.
  rename H7 into ev.
  rename v0 into res.
  rename H8 into res_equ.
  rename H9 into mem_equ.

The generated hypotheses have become

res : val
ev : eval_expr env c_env tmp_env m1
       (Ebinop Oadd
               (Etempvar _x tint)
               (Etempvar _y tint)
               tint)
       res
res_equ : sem_cast res tint tint m1 = Some v'
mem_equ : Memory.Mem.free_list m1
                               (blocks_of_env env c_env)
            = Some m'0

Our understanding of these hypotheses is the following

The expression _x + _y is evaluated using the c_env environment (and we know thanks to binding the value of _x and _y), and its result is stored in res
res is cast into a tint value, and acts as the result of f_add

The Hstar hypothesis is now interesting

Hstar : Smallstep.star
          step2 env
          (Returnstate v' Kstop m'0) t0
          (Returnstate (Vint z) Kstop m')

It is clear that we are at the end of the execution of f_add (even if Coq generates two subgoals, the second one is not relevant and easy to discard).

  smart_inv Hstar; [| smart_inv H ].

We are making good progress here, and we can focus our attention on the ev hypothesis, which concerns the evaluation of the _x + _y expression. We can simplify it a bit further.

  smart_inv ev; [| smart_inv H].
  rename H4 into fetch_x.
  rename H5 into fetch_y.
  rename H6 into add_op.

In a short-term, the hypotheses fetch_x and fetch_y are the most important.

fetch_x : eval_expr env c_env tmp_env m1 (Etempvar _x tint) v1
fetch_y : eval_expr env c_env tmp_env m1 (Etempvar _y tint) v2

The current challenge we face is to prove that we know their value. At this point, we can have a look at f_entry. This is starting to look familiar: smart_inv, then renaming, etc.

  smart_inv f_entry.
  clear H.
  clear H0.
  clear H1.
  smart_inv H3; subst.
  rename H2 into allocs.

We are almost done. Let’s simplify as much as possible fetch_x and fetch_y. Each time, the smart_inv tactic generates two subgoals, but only the first one is relevant. The second one is not, and can be discarded.

  smart_inv fetch_x; [| inversion H].
  smart_inv H2.
  smart_inv fetch_y; [| inversion H].
  smart_inv H2.

We now know the values of the operands of add. The two relevant hypotheses that we need to consider next are add_op and res_equ. They are easy to read.

add_op : sem_binarith
           (fun (_ : signedness) (n1 n2 : Integers.int)
              => Some (Vint (add n1 n2)))
           (fun (_ : signedness) (n1 n2 : int64)
              => Some (Vlong (Int64.add n1 n2)))
           (fun n1 n2 : Floats.float
              => Some (Vfloat (Floats.Float.add n1 n2)))
           (fun n1 n2 : Floats.float32
              => Some (Vsingle (Floats.Float32.add n1 n2)))
           v1 tint v2 tint m1 = Some res

add_op is the addition of Vint x and Vint y, and its result is res.
```
res_equ : sem_cast res tint tint m1 = Some (Vint z)
```
res_equ is the equation which says that the result of f_add is res, after it has been cast as a tint value.

We can simplify add_op and res_equ, and this allows us to conclude.

  smart_inv add_op.
  smart_inv res_equ.
  reflexivity.
Qed.

Conclusion

The definitions of Clight are straightforward, and the CompCert documentation is very pleasant to read. Understanding Clight and its semantics can be very interesting if you are working on a language that you want to translate into machine code. However, proving some functional properties of a given C snippet using only CompCert can quickly become cumbersome. From this perspective, the VST project is very interesting, as its main purpose is to provide tools to reason about Clight programs more easily.